Average Error: 46.7 → 0.0
Time: 2.4s
Precision: 64
\[i \gt 0.0\]
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 204.76020695153215:\\ \;\;\;\;\frac{i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \frac{i}{2 \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]
\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}
↓
\begin{array}{l}
\mathbf{if}\;i \le 204.76020695153215:\\
\;\;\;\;\frac{i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \frac{i}{2 \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\

\end{array}
double f(double i) {
        double r77225 = i;
        double r77226 = r77225 * r77225;
        double r77227 = r77226 * r77226;
        double r77228 = 2.0;
        double r77229 = r77228 * r77225;
        double r77230 = r77229 * r77229;
        double r77231 = r77227 / r77230;
        double r77232 = 1.0;
        double r77233 = r77230 - r77232;
        double r77234 = r77231 / r77233;
        return r77234;
}
↓
double f(double i) {
        double r77235 = i;
        double r77236 = 204.76020695153215;
        bool r77237 = r77235 <= r77236;
        double r77238 = 2.0;
        double r77239 = r77238 * r77235;
        double r77240 = r77239 * r77239;
        double r77241 = 1.0;
        double r77242 = r77240 - r77241;
        double r77243 = r77235 / r77242;
        double r77244 = r77238 * r77238;
        double r77245 = r77235 / r77244;
        double r77246 = r77243 * r77245;
        double r77247 = 0.00390625;
        double r77248 = 1.0;
        double r77249 = 4.0;
        double r77250 = pow(r77235, r77249);
        double r77251 = r77248 / r77250;
        double r77252 = r77247 * r77251;
        double r77253 = 0.015625;
        double r77254 = 2.0;
        double r77255 = pow(r77235, r77254);
        double r77256 = r77248 / r77255;
        double r77257 = r77253 * r77256;
        double r77258 = 0.0625;
        double r77259 = r77257 + r77258;
        double r77260 = r77252 + r77259;
        double r77261 = r77237 ? r77246 : r77260;
        return r77261;
}

Error

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if i < 204.76020695153215

    1. Initial program 45.1

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]
    3. Using strategy rm

    4. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \frac{i}{2 \cdot 2}}\]

    if 204.76020695153215 < i

    1. Initial program 48.3

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1}\]

    2. Simplified32.7

      \[\leadsto \color{blue}{\frac{i \cdot i}{\left(\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1\right) \cdot \left(2 \cdot 2\right)}}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 204.76020695153215:\\ \;\;\;\;\frac{i}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1} \cdot \frac{i}{2 \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.00390625 \cdot \frac{1}{{i}^{4}} + \left(0.015625 \cdot \frac{1}{{i}^{2}} + 0.0625\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :precision binary64
  :pre (and (> i 0.0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1)))